I suspect that most people would agree with Jeff Ketland that a statement
like
(*) There is a property P such that for all y, y has P iff phi(y)
cannot easily be seen as tautological or logically true.
It doesn't follow, however, that second-order comprehension isn't a
logical truth, because (*) is not the only possible interpretation of SO
comprehension. If we assume, as several messages have suggested, that
what prevents (*) from being a logical truth is that it makes an existence
claim, then the questionis whether we can find interpretations of SO
comprehension that are free of ontological implications.
As Jeff says, if we limit ourselves to monadic comprehension then Boolos
has an interesting idea about this, using plural quantification. Monadic
SO comprehension can be read as saying that
(**) there are some things such that for all y, y is one of them iff phi(y)
Question: do list members think (there's any chance that) (**) is to be
regarded as logically true?
Unfortunately the only clear way to boot (**) up to full comprehension
involves an ontological assumption, viz. that whenever there are objects
x1....xn, there's a further thing that's their n-tuple. Given that
assumption we can construe n-adic SO quantification as monadic plural
quantification over n-tuples. But if we're looking for a logically true
reading of comprehension then we'd better look elsewhere; logic alone
can't give you n-tuples.
The problem is to find an ontologically innocent take on polyadic SO
comprehension. Here's a proposal, which I admit sounds somewhat weird,
that seems to me sometimes to do the trick. It exploits the colloquial
device, not of plural quantification, but non-nominal quantification (as
in: "he did it somehow," "however he did it, she did it too.")
(***) things relate somehow suchthat x1....xn are so related iff
phi(x1,....,xn).
I've got three questions about this. First, colloquial weirdness aside,
does (***)
capture the (or a) meaning of polyadic SO comprehension? Second, is (***)
ontologically innocent, in the way that (*) seems to be? Third, is (***)
logically true?
Steve Yablo
MIT philosophy